Polytope of Type {4,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,108}*864c
if this polytope has a name.
Group : SmallGroup(864,613)
Rank : 3
Schlafli Type : {4,108}
Number of vertices, edges, etc : 4, 216, 108
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,108,2} of size 1728
Vertex Figure Of :
{2,4,108} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,54}*432b
3-fold quotients : {4,36}*288c
4-fold quotients : {4,27}*216
6-fold quotients : {4,18}*144b
9-fold quotients : {4,12}*96c
12-fold quotients : {4,9}*72
18-fold quotients : {4,6}*48c
36-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,108}*1728b
Permutation Representation (GAP) :
s0 := ( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)( 8,114)
( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)
( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214)(217,327)(218,328)(219,325)(220,326)
(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)(228,334)
(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)(236,342)
(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)(244,350)
(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)(252,358)
(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)(260,366)
(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)(268,374)
(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)(276,382)
(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)(284,390)
(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)(292,398)
(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)(300,406)
(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)(308,414)
(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)(316,422)
(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)(324,430);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)( 43,102)
( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)
( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)
( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)( 67, 78)
( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)
(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)(146,215)
(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)(154,207)
(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)(162,199)
(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)(170,191)
(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)(178,183)
(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)(222,335)
(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)(230,359)
(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)(238,351)
(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)(246,343)
(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)(254,431)
(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)(262,423)
(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)(270,415)
(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)(278,407)
(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)(286,399)
(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)(294,391)
(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)(302,383)
(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)(310,375)
(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)(318,367)
(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);;
s2 := ( 1,289)( 2,292)( 3,291)( 4,290)( 5,297)( 6,300)( 7,299)( 8,298)
( 9,293)( 10,296)( 11,295)( 12,294)( 13,321)( 14,324)( 15,323)( 16,322)
( 17,317)( 18,320)( 19,319)( 20,318)( 21,313)( 22,316)( 23,315)( 24,314)
( 25,309)( 26,312)( 27,311)( 28,310)( 29,305)( 30,308)( 31,307)( 32,306)
( 33,301)( 34,304)( 35,303)( 36,302)( 37,253)( 38,256)( 39,255)( 40,254)
( 41,261)( 42,264)( 43,263)( 44,262)( 45,257)( 46,260)( 47,259)( 48,258)
( 49,285)( 50,288)( 51,287)( 52,286)( 53,281)( 54,284)( 55,283)( 56,282)
( 57,277)( 58,280)( 59,279)( 60,278)( 61,273)( 62,276)( 63,275)( 64,274)
( 65,269)( 66,272)( 67,271)( 68,270)( 69,265)( 70,268)( 71,267)( 72,266)
( 73,217)( 74,220)( 75,219)( 76,218)( 77,225)( 78,228)( 79,227)( 80,226)
( 81,221)( 82,224)( 83,223)( 84,222)( 85,249)( 86,252)( 87,251)( 88,250)
( 89,245)( 90,248)( 91,247)( 92,246)( 93,241)( 94,244)( 95,243)( 96,242)
( 97,237)( 98,240)( 99,239)(100,238)(101,233)(102,236)(103,235)(104,234)
(105,229)(106,232)(107,231)(108,230)(109,397)(110,400)(111,399)(112,398)
(113,405)(114,408)(115,407)(116,406)(117,401)(118,404)(119,403)(120,402)
(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)(128,426)
(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)(136,418)
(137,413)(138,416)(139,415)(140,414)(141,409)(142,412)(143,411)(144,410)
(145,361)(146,364)(147,363)(148,362)(149,369)(150,372)(151,371)(152,370)
(153,365)(154,368)(155,367)(156,366)(157,393)(158,396)(159,395)(160,394)
(161,389)(162,392)(163,391)(164,390)(165,385)(166,388)(167,387)(168,386)
(169,381)(170,384)(171,383)(172,382)(173,377)(174,380)(175,379)(176,378)
(177,373)(178,376)(179,375)(180,374)(181,325)(182,328)(183,327)(184,326)
(185,333)(186,336)(187,335)(188,334)(189,329)(190,332)(191,331)(192,330)
(193,357)(194,360)(195,359)(196,358)(197,353)(198,356)(199,355)(200,354)
(201,349)(202,352)(203,351)(204,350)(205,345)(206,348)(207,347)(208,346)
(209,341)(210,344)(211,343)(212,342)(213,337)(214,340)(215,339)(216,338);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)
( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)
( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)
( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)
( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)
( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)
( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)
( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214)(217,327)(218,328)(219,325)
(220,326)(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)
(228,334)(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)
(236,342)(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)
(244,350)(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)
(252,358)(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)
(260,366)(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)
(268,374)(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)
(276,382)(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)
(284,390)(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)
(292,398)(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)
(300,406)(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)
(308,414)(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)
(316,422)(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)
(324,430);
s1 := Sym(432)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)
( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)
( 23, 26)( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)
( 43,102)( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)
( 51, 94)( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)
( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)
(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)
(146,215)(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)
(154,207)(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)
(162,199)(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)
(170,191)(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)
(178,183)(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)
(222,335)(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)
(230,359)(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)
(238,351)(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)
(246,343)(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)
(254,431)(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)
(262,423)(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)
(270,415)(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)
(278,407)(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)
(286,399)(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)
(294,391)(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)
(302,383)(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)
(310,375)(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)
(318,367)(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);
s2 := Sym(432)!( 1,289)( 2,292)( 3,291)( 4,290)( 5,297)( 6,300)( 7,299)
( 8,298)( 9,293)( 10,296)( 11,295)( 12,294)( 13,321)( 14,324)( 15,323)
( 16,322)( 17,317)( 18,320)( 19,319)( 20,318)( 21,313)( 22,316)( 23,315)
( 24,314)( 25,309)( 26,312)( 27,311)( 28,310)( 29,305)( 30,308)( 31,307)
( 32,306)( 33,301)( 34,304)( 35,303)( 36,302)( 37,253)( 38,256)( 39,255)
( 40,254)( 41,261)( 42,264)( 43,263)( 44,262)( 45,257)( 46,260)( 47,259)
( 48,258)( 49,285)( 50,288)( 51,287)( 52,286)( 53,281)( 54,284)( 55,283)
( 56,282)( 57,277)( 58,280)( 59,279)( 60,278)( 61,273)( 62,276)( 63,275)
( 64,274)( 65,269)( 66,272)( 67,271)( 68,270)( 69,265)( 70,268)( 71,267)
( 72,266)( 73,217)( 74,220)( 75,219)( 76,218)( 77,225)( 78,228)( 79,227)
( 80,226)( 81,221)( 82,224)( 83,223)( 84,222)( 85,249)( 86,252)( 87,251)
( 88,250)( 89,245)( 90,248)( 91,247)( 92,246)( 93,241)( 94,244)( 95,243)
( 96,242)( 97,237)( 98,240)( 99,239)(100,238)(101,233)(102,236)(103,235)
(104,234)(105,229)(106,232)(107,231)(108,230)(109,397)(110,400)(111,399)
(112,398)(113,405)(114,408)(115,407)(116,406)(117,401)(118,404)(119,403)
(120,402)(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)
(128,426)(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)
(136,418)(137,413)(138,416)(139,415)(140,414)(141,409)(142,412)(143,411)
(144,410)(145,361)(146,364)(147,363)(148,362)(149,369)(150,372)(151,371)
(152,370)(153,365)(154,368)(155,367)(156,366)(157,393)(158,396)(159,395)
(160,394)(161,389)(162,392)(163,391)(164,390)(165,385)(166,388)(167,387)
(168,386)(169,381)(170,384)(171,383)(172,382)(173,377)(174,380)(175,379)
(176,378)(177,373)(178,376)(179,375)(180,374)(181,325)(182,328)(183,327)
(184,326)(185,333)(186,336)(187,335)(188,334)(189,329)(190,332)(191,331)
(192,330)(193,357)(194,360)(195,359)(196,358)(197,353)(198,356)(199,355)
(200,354)(201,349)(202,352)(203,351)(204,350)(205,345)(206,348)(207,347)
(208,346)(209,341)(210,344)(211,343)(212,342)(213,337)(214,340)(215,339)
(216,338);
poly := sub<Sym(432)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope